The Quantum-to-Classical Correspondence in Statistical Mechanical Systems

Abstract

Quantum-to-classical correspondence of systems is crucial for reducing the complexity of certain quantum problems. We study a variant of one of the most well-understood models, a “mixed” two-dimensional Ising model, showing that it can be understood as a fermionic system. We show that the partition function, $Z$, and critical behavior of the mixed 2-dimensional classical Ising model are the same as that of the 1-dimensional quantum Ising model. We reformulate our Ising models in terms of percolation to show their equivalence. Our primary goal is to demonstrate the methods of solving quantum lattice systems and emphasize how they can be equivalent to classical models.